\[\int_{0}^{\infty}e^{-t}t^{b-1-\frac{1}{2}\nu}{\mathbf{M}}\left(a,b,t\right)J_{%
\nu}\left(2\sqrt{xt}\right)\,\mathrm{d}t=x^{-a+\frac{1}{2}\nu}e^{-x}{\mathbf{M%
}}\left(\nu-b+1,\nu-a+1,x\right),\]